Submerged Landau jet I am reading Landau & Lifshitz's Fluid Mechanics. On page 81, section 23, it reads

Determine the flow in a jet emerging from the end of a narrow tube into an infinite space filled with the fluid - the submerged jet. We take spherical polar coordinates $r,\theta,\phi$, with the polar axis in the direction of the jet at its point of emergence, and with this point as origin. The flow is symmetrical about the polar axis, so that $v_\phi=0$ and $v_\theta,v_r$ are functions of $r$ and $\theta$ only. The same total momentum flux (the "momentum of the jet") must pass through any closed surface surrounding the origin (in particular, through an infinitely distant surface). For this to be so, the velocity must be inversely proportional to $r$, so that $$v_r=F(\theta)/r, v_\theta=f(\theta)/r, \tag{23.16}$$ where $F$ and $f$ are some functions of $\theta$ only.

I don't know how $(23.16)$ is derived. Can someone explain it?
 A: 

The same total momentum flux (the "momentum of the jet") must pass through any closed surface surrounding the origin (in particular, through an infinitely distant surface). For this to be so, the velocity must be inversely proportional to $r$, so that $$v_r=F(\theta)/r, v_\theta=f(\theta)/r, \tag{23.16}$$ where $F$ and $f$ are some functions of $\theta$ only.


(Emphasis added)

I don't know how $(23.16)$ is derived. Can someone explain it?

You need to use the definition of convective momentum flux density. There are two factors of $v$ in this momentum flux density: The flux of momentum across a surface of area $\vec {dA}$ is
$$
\rho \vec v \vec v \cdot \vec{dA}\;.
$$
Or, for a far distant spherical surface of radius $r$:
$$
\int d\Omega \rho\vec v v_r r^2\;.
$$
Regardless of if I look at the $r$ component or the $\theta$ component of the above, I am going to need
$$
v_r\vec v \sim \frac{1}{r^2}\;,
$$
Or, looking at the $r$ component:
$$
v_r \sim \frac{1}{r}\;,
$$
which therefore also implies for the $\theta$ component:
$$
v_\theta \sim \frac{1}{r}\;,
$$
and which justifies the functional form proposed by Landua.
A: Since the problem has an axial symmetry then he rid off the $\phi$ angle, i.e. no dependence on the azimuthal angle $\phi$ hints to the absence of this variable in the solution in the function $F$ or $f$.
As to the $r^{-1}$ the spherical diverting waves loss their magnitude (or Intensity in other words) by this low because it is required by the conversation of momentum (one of the Newton's law).
Therefore his assumption is to find the velocities in the such forms.
